Apparatus and methods for gaussian boson sampling

ABSTRACT

An apparatus includes a light source to provide a plurality of input optical modes in a squeezed state. The apparatus also includes a network of interconnected reconfigurable beam splitters (RBSs) configured to perform a unitary transformation of the plurality of input optical modes to generate a plurality of output optical modes. An array of photon counting detectors is in optical communication with the network of interconnected RBSs and configured to measure the number of photons in each mode of the plurality of the output optical modes after the unitary transformation. The apparatus also includes a controller operatively coupled to the light source and the network of interconnected RBSs. The controller is configured to control at least one of the squeezing factor of the squeezed state of light, the angle of the unitary transformation, or the phase of the unitary transformation.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of, and priority to, U.S.Provisional Patent Application No. 62/851,312, filed May 22, 2019 andtitled “Apparatus and Methods for Gaussian Boson Sampling,” the entirecontent of which is incorporated by reference herein in its entirety.

FIELD

One or more embodiments relate to Gaussian Boson sampling.

BACKGROUND

Algorithms employing universal quantum computation often promisesubstantially faster computational speeds compared to their classicalcounterparts. To date, however, no known hardware is capable of runningthese algorithms on large-scale problems, much less in a fault-tolerantmanner.

SUMMARY

Some embodiments described herein relate generally to Gaussian Bosonsampling, and, in particular, to performing Gaussian Boson sampling on aphotonic platform. In some embodiments, an apparatus includes a lightsource configured to provide a plurality of input optical modes in asqueezed state of light. The apparatus also includes a network ofinterconnected reconfigurable beam splitters (RBSs) in opticalcommunication with the light source. The network of interconnected RBSsis configured to perform a unitary transformation on the plurality ofinput optical modes to generate a plurality of output optical modes. Anarray of photon counting detectors is in optical communication with thenetwork of interconnected RBSs and configured to measure the number ofphotons in each mode of the plurality of the output optical modes afterthe unitary transformation. The apparatus also includes a controlleroperatively coupled to the light source and the network ofinterconnected RBSs. The controller is configured to control at leastone of the squeezing factor of the squeezed state of light, the angle ofthe unitary transformation, or the phase of the unitary transformation.

BRIEF DESCRIPTION OF THE DRAWINGS

The drawings primarily are for illustration purposes and are notintended to limit the scope of the subject matter described herein. Thedrawings are not necessarily to scale; in some instances, variousaspects of the disclosed subject matter disclosed herein may be shownexaggerated or enlarged in the drawings to facilitate an understandingof different features. In the drawings, like reference charactersgenerally refer to like features (e.g., functionally similar and/orstructurally similar elements).

FIG. 1 shows a schematic of an apparatus for Gaussian Boson sampling(GBS), according to an embodiment.

FIG. 2 shows a schematic of a GBS system using spatial encoding,according to an embodiment.

FIG. 3 shows a schematic of a quantum processing unit (QPU) implementedon an integrated photonic circuit to perform GBS, according to anembodiment.

FIGS. 4A-4B show schematics of power division modules that can be usedin the QPU shown in FIG. 3, according to an embodiment,

FIG. 5 shows a schematic of a displacement module that can be used toproduce displaced squeezed light for GBS, according to an embodiment.

FIG. 6 shows a block diagram to illustrate a method of operating a GBSdevice, according to an embodiment.

FIG. 7 shows a schematic of a GBS system that provide user access to thesetting of the quantum hardware, according to an embodiment,

FIG. 8 shows a block diagram illustrating a method of solving graphproblems using GBS devices, according to an embodiment.

DETAILED DESCRIPTION

In view of the challenges in realizing universal quantum computation,alternative, non-universal quantum computation models are currentlyunder active research. One such model is Gaussian boson sampling (GBS).A GBS system prepares input squeezed vacuum or squeezed coherent statesof a set of quantum harmonic oscillators (e.g., optical field modes) inM modes, inputs them to an N-mode linear optical interferometer, anddetects the number of photons in each of the N output modes of theinterferometer. In other words, a GBS system samples from theprobability distribution of the output in the photon number basis.

FIG. 1 shows a schematic of an apparatus 100 for GBS, according to anembodiment. The apparatus 100 includes an input state generator 110 toprepare the desired input states of light (e.g., optical modes),followed by a linear interferometer 120 to perform a unitarytransformation on the input states of light received from the inputstate generator 110. A detector 130 is employed to measure the output ofthe linear interferometer 120 (e.g., output optical modes, also referredto as transformed optical modes). The combination of the input stategenerator 110 and the linear interferometer 120 is also referred to as aquantum processing unit (QPU) 150, and components within the dashed lineis also referred to as quantum hardware (QH) 160.

In addition to the QH 160, the apparatus 100 also includes an inputcontroller 115 to control the operation of the input state generator110, an interferometer controller 125 to control the operation of thelinear interferometer 120, and a detection controller 135 to control theoperation of the detector 130. Furthermore, a master controller 140 isemployed to coordinate and/or control of the operations of the threecontrollers 115, 125, and 135. The controllers 115, 125, and 135 caninclude one or more field-programmable gate arrays (FPGAs),application-specific integrated circuits (ASICs), graphics processingunits (GPUs) and/or central processing units (CPUs). The use of FPGAs,ASICs, GPUs and/or CPUs can result in increased operational speed,reliability and flexibility, as well as lower monetary cost, as comparedwith customized integrated circuits.

In some embodiments, the controllers 115, 125, and 135 can include smallcomputers running drivers to perform control logic and actuate thevarious subsystem hardware elements. The master controller 140 caninclude a computer that co-ordinates the control logic of the subcontrollers 115, 125, and 135. For example, the master controller 140can be configured to receive instructions from a user and then send thereceived instructions to one or more specific sub controllers.

In some embodiments, the input state generator 110 is configured togenerate the input states as optical beams or optical pulses (e.g.,squeezed light). In these embodiments, the input state generator 110 isalso referred to as a light source 110. For example, the input stategenerator 110 converts laser light and creates, via a nonlinear opticalelement, a total number M of squeezed and/or displaced squeezed statesof light, which can be represented by

$\prod\limits_{i = 1}^{M}{{D_{i}(\alpha)}{S_{i}( r_{i} )}{{{{vac} >},{{{where}\mspace{14mu}{D_{i}(\alpha)}} = {\exp( {{\alpha\; a_{i}^{+}} - {\alpha^{*}a_{i}}} )}},{{{and}\mspace{14mu}{S_{i}(r)}} = {\exp( {{\frac{r}{2}a_{i}a_{i}} - {\frac{r^{*}}{2}a_{i}^{+}a_{i}^{+}}} )}},}}}$

i is the mode number, r_(i) is the squeezing factor, and a_(i) is anannihilation operator. In some embodiments, the squeezing factor anddisplacement in each mode can be independently adjusted so as to encodedifferent applications into the apparatus 100. In some embodiments, thedisplacement variable α can also be different for each mode, i.e., eachmode i has a corresponding displacement α₄ in above equations.

The linear interferometer 120 is configured to perform a unitaryoperation on up to N input modes, and therefore transforms the inputmode operator a_(i) ⁺, with a photon in mode i represented by a_(i)⁺|vac>, to the output mode operator Σ_(j=1) ^(N)U_(ij)a_(j) ⁺, where Uis the unitary operator. Different unitary operators U_(ij) can beprogrammed into the linear interferometer 120 for each run of the QH 160in the apparatus 100.

The detector 130 in the apparatus 100 has photon counting capability,i.e., the detector 130 is configured to measure the number of photons ineach output mode of the linear interferometer 120. Without being boundby any particular theory or mode of operation, each output mode of thelinear interferometer 120 can be represented by the projection-valuedmeasure elements

$\prod\limits_{i}^{N}{= {\frac{1}{N_{1}}( a_{i}^{+} )^{N}{{{vac} > < {vac}}}{( a_{i} )^{N}.}}}$

The modes i generated by the input state generator 110 (i.e., the stateencoding) can correspond to any appropriate set of quantum harmonicoscillator modes. Optical field modes are used here as illustratingexamples. In general, an optical field mode can be characterized byfrequency/time, polarization, and spatial distribution. Therefore,various encoding schemes can be employed to prepare appropriate opticalfield modes for the apparatus 100 to perform GBS.

In some embodiments, the input state generator 110 can be configured togenerate optical modes using spatial encoding. In these embodiments,different optical modes are characterized by transverse fielddistributions that do not overlap in space. For example, differentoptical modes can include separate light beams, which can eitherpropagate in free space or in separate optical waveguides.

In some embodiments, the apparatus 100 can include multiple waveguidesto propagate optical modes in spatial encoding. For example, the linearinterferometer 120 can include M input ports to receive the M inputoptical modes, and each port can be coupled to an input waveguide thatpropagates a distinct input optical mode. In addition, the output modesof the linear interferometer 120 can also be transmitted to the detector130 via waveguides. In this case, the detector 130 can include an arrayof photon counting detectors, each of which is coupled to acorresponding output port of the linear interferometer 120 to measurethe number of photons in the output optical mode delivered from theoutput port.

In some embodiments, the input state generator 110 can be configured togenerate optical modes using time encoding. In these embodiments,different optical modes are characterized by longitudinal distributionsthat do not overlap in space. For example, each optical mode can includea well-defined optical pulse in a single longitudinal path.

In some embodiments, the input state generator 110 can be configured togenerate optical modes using frequency encoding. In these embodiments,different optical modes are characterized by frequency distributionsthat do not overlap in frequency, i.e., each optical mode has a distinctfrequency or wavelength. In some embodiments, the input state generator110 can employ more than one encoding scheme to prepare the inputoptical modes, i.e., using a hybrid scheme.

FIG. 2 shows a schematic of a GBS system 200 using spatial encoding,according to an embodiment. The system 200 includes a server 210 fornetworking the system to a user terminal 201. The user terminal 201 caninclude any user hardware for accessing the system 200. The access canbe realized locally or by internet connection. The system 200 alsoincludes a master unit 240 for controlling the system 200, receivinguser instructions, and/or serving status information and computationresults to users.

In the system 200, a QPU 250 is used to perform the GBS. The QPU 250 canbe substantially similar to the QPU 150 shown in FIG. 1 and describedabove (e.g., including an input state generator and a linearinterferometer). In some embodiments, the QPU 250 can be configured as aphotonic chip (see more details below with reference to FIGS. 3-4), AQPU driver 252 is operatively coupled to the QPU 250 for configuring,programming, and operating the elements of the QPU 250. For example, theQPU driver 252 can include a suitable voltage/current driver for on-chipelectrical elements (e.g., phase shifters) that are included in the QPU250. In addition, a locking loop controller 254 is employed for anyactive feedback systems associated with the QPU 250, such as lockingloops for resonant devices. The QPU driver 252 and/or the locking loopcontroller 254 can include one or more FPGAs, ASICs, GPUs and/or CPUs.

The QPU 250 is also operatively coupled to a QPU environment controller255, which is configured to control the mechanical, thermal, and/oroptical environment of the QPU 250. For example, the QPU environmentcontroller 255 can be configured to ensure that the QPU 250 is operatingwithin a stable environment (e.g., low level of temperature fluctuation,etc.). The QPU environment controller 255 can include one or more FPGAs,ASICs, GPUs and/or CPUs. One or more monitor photodiodes 260 areoperatively coupled to the QPU 250 and the QPU environment controller255 for assessing the stability of the QPU elements. The monitorphotodiodes 260 can be configured to measure operation parameters (e.g.,optical output) of the QPU 250 and provide the measurement to the QPUenvironment controller 255 so as to allow the QPU environment controller255 to generate appropriate control signals.

The system 200 also includes a pump source 220 to provide pump lightbeams for the input state generator within the QPU 250 to generate theappropriate input states. A pump controller 225 is operatively coupledto the pump source 220 to control the operation of the pump source 220.The pump controller 225 can include one or more FPGAs, ASICs, GPUsand/or CPUs. In some embodiments, the pump source 220 can include alaser.

The output of the QPU 250 is monitored by a detector array 230, whichincludes a set of photon counting detectors, such asphoton-number-resolving transition edge sensors, superconductingnanowire photon counting detectors, and/or avalanche photodiodes. Thedetector array 230 is controlled by a detection controller 235. Forexample, the detection controller 235 can adjust the biasing voltage ofthe detector array 230 so as to improve the detection efficiency. Thedetection controller 235 can include one or more FPGAs, ASICs, GPUsand/or CPUs. In some embodiments, an optional cryogenic controller 234can be employed to control the cryogenic system (not shown in FIG. 2)that supports the operation of the detector array 230. A dataacquisition system (DAQ) 232 is operatively coupled to the detectorarray 230 to translate the output signals of the detector array 230 intoa series of integers representing photon number readout, which can thenbe reported to users (e.g., via the user terminal 201).

FIG. 3 shows a schematic of a quantum processing unit (QPU) 300implemented on an integrated photonic circuit to perform GBS, accordingto an embodiment. The QPU 300 includes an input state generator 310 andan interferometer 320 that are fabricated in a substrate 305 (alsoreferred to as a platform 305). The substrate can include one or more ofsilicon, silicon oxide (e.g., silicon dioxide), silicon nitride, etc.The input state generator 310 includes a power division module (PDM)312, a squeezer 314, a filter 316, and an optional displacement module318 (to generate displaced squeezed light). The interferometer 320includes input ports 322, a network of RBSs 325 to perform unitarytransformation, and output ports 328.

The QPU 300 also includes a QPU driver 340 operatively coupled to thePDM 312, the squeezer 314, the filter 316, the displacement module 318,and the network of RBSs 325 to control their operations. In addition,each of the above elements has a corresponding monitor (e.g.,photodetectors and electronics) for monitoring the optical statusoutputs of that element. More specifically, the PDM 312 is coupled to aPDM monitor 313, the squeezer 314 is coupled to a squeezer monitor 315,the filter 316 is coupled to a filter monitor 317, the displacementmodule 318 is coupled to a displacement monitor 319, and the network ofRBSs 325 is coupled to an interferometer monitor 324. All the monitors313, 315, 317, 319, and 324 are also operatively coupled to the QPUdriver 340. In FIG. 3, thin lines (e.g., between each monitor and itscorresponding element) indicate optical signals, and thick lines witharrows (e.g., between the QPU driver 340 and components in the inputstate generator 310 and the interferometer 320) indicate electricalsignals.

The platform 305 can be selected for low optical loss and high indexcontrast for high confinement of optical modes. Therefore, the resultingbuilding blocks (e.g., waveguides) can be used to construct large-depthoptical circuits or circuit elements (e.g., PDM 312, squeezer 314,displacement module 318, or connections between these elements, etc.)where optical modes can propagate a long distance. In some embodiments,the platform 305 includes a silicon-on-insulator wafer that supportsoptical waveguide structures and integrated photonic components.

In addition to good passive performance, the waveguide material can alsopossess a high second-order or third-order nonlinear optical response soas to induce efficient parametric fluorescence, which can be employed bythe input state generator 310 to generate squeezed light, in someembodiments, the waveguide material can include silicon nitride, siliconnanowire, lithium niobate, and aluminum nitride, among others.

In some embodiments, the QPU 300 can be fabricated using a fabricationplatform involving multiple wave guiding layers, which can offerenhanced flexibility in design. For example, nonlinear opticalcomponents (e.g., the squeezer 314) can be fabricated on a siliconnitride waveguide layer, and the interferometer 320 can be implementedon a silicon layer. In these embodiments, interlayer couplers can beemployed to allow the transfer of light between different layers.

In some embodiments, it can be beneficial to have tunability andprogrammability in some active components in the QPU 300 (e.g., theinterferometer 320). The tunability and programmability can be achievedvia electrical inputs from the QPU driver 340. In some embodiments,thermo-optical, electro-optical phase shifters, or any other appropriatetype of phase shifters can be integrated alongside the optical waveguidestructures so as to realize tunability and programmability.

In the QPU 300, the input state generator 310 is configured to createsqueezed and/or displaced squeezed states of light for input into thelinear optical interferometer 320. As an input, the state generator 310converts coherent laser light to drive a nonlinear optical process toproduce squeezing and to perform coherent displacements. Generally,squeezers and displacers (e.g., 314 and 318) are pumped by mutuallycoherent beams, and the PDM 312 is employed to take one or more mutuallycoherent laser input beams and produce multiple phase locked andmutually coherent output laser beams to drive the squeezer anddisplacers. The output of the PDM 312 is then directed to the squeezer314 and displacement module 318.

In some embodiments, the displacement module 318 can be optional, i.e.,the input optical modes into the interferometer 320 does not includedisplacement. In these embodiments, the PDM 312 can be configured toprovide an array of high power laser beams to pump the squeezer 314. Insome embodiments, the input optical modes into the interferometer 320includes displaced squeezed light.

In these embodiments, the PDM 312 is configured to provide at least oneadditional output at lower power for use as a coherent displacementbeam. Generally, the wavelength of the pump light beams (i.e., pumpwavelength) that drive the nonlinear process to create the squeezedlight is different from the wavelength of the resulting squeezed light(i.e., squeezed light wavelength). Accordingly, the wavelength of thedisplacement beam (i.e., displacement wavelength) is also different fromthe pump wavelength. In addition, for many mode systems (i.e., large-Msystem, where M is the number of optical modes), the high optical powerdemands may make it challenging to receive the entire required quantityof pump light power in a single channel.

In some embodiments, the QPU 300 can include beam splitters to dividethe pump beam into multiple separate pump beams (e.g., M pump beams) andcan monitor them to provide information on the relative phase betweenthe set of divided input pump beams. The phase information is then madeavailable to the QPU driver 340 to feed back on the relevant input pumpsand lock their relative phases. As used herein, a beam splitter (BS)refers to a component that transfers power from two input modes to twooutput modes via, e.g., a directional coupler or a multi-modeinterferometer.

In some embodiments, the beam splitter used can include a reconfigurablebeam splitter (RBS), which has tunable power splitting ratios. Forexample, an RBS can include a Mach-Zehnder interferometer formed fromtwo subsequent beam splitters, with a tunable phase shifter (TPS) in oneintermediate arm to provide controllable transmission ratios betweenfour ports. As used herein, a TPS is an element placed adjacent to or inthe path of a waveguide that provides a controllable optical phasedelay. A TPS can operate thereto-optically or electro-optically (or byany other appropriate mechanism). In general, a TPS can be tuned by aninput voltage applied to electrical contacts on the photonic chip thatincludes the TPS.

In some embodiments, the phase information can be acquired by tappingoff a small fraction of each input pump beam upon entry to the PDM 312with an on-chip beam splitter (e.g., a directional coupler). Each tappedfraction can be interfered with the neighboring tapped fraction viaanother beam splitter, and the resultant optical outputs of that beamsplitter are made available to photodetectors. The information from thesignals incident on these photodetectors can be used by the QPU driver340 to feedback and lock the relative pump input phases. In someembodiments, the feedback is actuated before the pump beams enter intothe QPU 300. In some embodiments, the feedback is actuated by TPS unitsintegrated in the PDM 312 via electrical signals from the QPU driver340.

It is also beneficial for the displacement beam (labelled as “DISPINPUT” in FIG. 3) to be phase stabilized relative to the pump beam(s).Therefore, the PDM 312 can include a monitor to probe this relativephase. Since the displacement wavelength is usually different from thepump wavelength, interfering the displacement beam with the pump beammay not be effective to extract the relative phase information. Toaddress this issue, a separate control beam carried in the same opticalchannels as the pump beams and displacement beam can be used to probethe relative phase of the displacement beam and pump beams. Thewavelength of the control beam is different from any other wavelengthsused (including the wavelengths of the pump beams, the displacementbeams, and the resulting squeezed light). In addition, the differencebetween the wavelength of the control beam and the wavelength of thesqueezed light can be sufficiently large to facilitate high-extinctionfiltering of the control beam before detection of the quantum light bythe photon counting detectors. Phase information is extracted byinterfering the control beams in different channels with one another,and measuring the resulting amplitude of the control beam afterinterference.

Following this phase locking layer of the PDM 312, the input pump beamsare divided to produce balanced output pump beams (e.g., M output pumpbeams) to drive the squeezer 314. In some embodiments, the division canbe realized by a set of beam splitter trees. In some embodiments, thedivision can be realized by an interferometer network. It is alsobeneficial to have independent control over the optical power of eachoutput pump beam (e.g., using RBSs) before the pump beam is directed tothe squeezer 314.

Depending on the nature of the squeezer 314, the output pump light beams(i.e., light beams that drive the squeezer 314) may be continuous,pulsed, monochromatic, or a combination of one or more continuous andone or more pulsed inputs at different wavelengths. For example, thesqueezer 314 may use a strong continuous drive beam and a weak pulsedbeam at another wavelength to generate squeezed light. More details ofthis scheme to generate squeezed light can be found in U.S. Pat. No.10,649,307, issued May 12, 2020 and titled “INTEGRATED DEVICES FORSQUEEZED LIGHT GENERATION,” which is incorporated herein it itsentirety. In this case, the PDM 312 allows bichromatic inputs to beappropriately routed based on the operation of the squeezer 314. If thetwo distinct wavelength components of the pump beams are carried inseparate channels before the QPU, a similar phase stabilizationmechanism can be incorporated to provide information on the requisiterelative phases.

In some embodiments, the PDM 312 is configured to operate withbichromatic pumps divided among multiple inputs and with a separatedisplacement beam and control beams in each input. In these embodiments,the PDM 312 operates as follows. First, the PDM 312 receives in separatechannels: (i) one or more pump inputs at a first wavelength (continuousor pulsed); (ii) one or more pump inputs at a second wavelength(continuous or pulsed); (iii) one input at a third wavelength in asuitable mode for coherent displacements. A control beam for probingphase displacements co-propagates in any input channel where thepropagating beam has a substantially different wavelength than theothers.

Then, the PDM 312 acquires information on the relative phases of theinputs. This information can be made available to an external driver(e.g., 340) via photodetection monitors (e.g., 313). The external driverthen modulates phase shifters (e.g., within the PDM 312, or prior to thePDM 312) to actuate phase shifts to correct for phase drifts between thevarious inputs and generate phase locked pump beams. The PDM 312 thendivides the phase-locked pumps into an appropriate number of outputswith variable power and directs them to the squeezer 314. The PDM 312also directs the phase-locked displacement beam to the displacementmodule 318.

FIG. 4A shows a schematic of a PDM 400 that can be used in the QPU 300shown in FIG. 3, according to an embodiment. The PDM 400 can divide aninput signal 401 arbitrarily into 2N output channels (i.e., P₁ to P_(2N)as illustrated in FIG. 4A). The PDM 400 includes 2N RBS units 401(1) to410(2N) and each RBS unit 410 includes an input port to receive an inputsignal and two output ports to deliver two output signals. In each RBS410, the first output port delivers a first output signal to anintegrated delay line (IDL) unit, which can delay the phase of the firstoutput signal by a desired amount. These delays can synchronize thearray of outputs of the PDM 400. The second output port of each RBS(e.g., 410(N)) delivers a second output signal to the input port of thenext RBS (e.g., 410(N+1)), i.e. the second output signal functions asthe input signal for the next RBS. For the last RBS (i.e., RBS_(2N)),the second output signal can be used as a test channel for calibration.

The transmission ratios of each RBS (i.e., T₁, T₂, . . . T_(2N)) can becontrolled by the corresponding phase shifters in each RBS (e.g., theTPS disposed on one arm of the RBS). These RBS units are configured toprovide the desired output powers in each channel, which can bedetermined by the configuration of the PDM 400. For example,odd-numbered output channels can carry pump light to drive the squeezer(e.g., 314 in FIG. 3), and even-numbered output channels can carry lightto perform the displacements (e.g., 318 in FIG. 3). That is, the(2j−1)th output for j=1, 2, 3, . . . , N corresponds to the channel thatcarries pump light for the squeezer element of the jthsqueezer-displacer, whereas the (2j)th output corresponds to thedisplacer input of the jth squeezer-displacer.

The RBS settings to generate a pre-determined sequence of output powerfractions can be determined as follows. Suppose that power P_(i) is atoutput channel i. Then the input power P_(in) to the PDM (neglectinglosses) can be written as:

P _(in)=Σ_(i) P _(i).  (1)

The fraction α_(i) of the total power that emerges from the output ofthe ith RBS is α_(i)=P_(i)/P₀; thus the output channel powertransmission ratio of the first RBS is T₁=α₁, and the output channelpower transmission ratio of the ith RBS (for i>1) is given by:

$\begin{matrix}{T_{i} = \frac{\alpha_{i}}{1 - {\sum\limits_{k = 1}^{i - 1}\alpha_{k}}}} & (2)\end{matrix}$

In some cases, these ratios (α_(i)) can be adjusted to account forpropagation losses. For example, the adjustment of the ratios (α_(i))can be determined during calibration of the PDM 400. Alternatively, theadjustment of the ratios (α_(i)) can be performed during operation ofthe PDM 400 in response to, for example, changes of environmentalparameters, such as temperature.

For pulsed operation, IDL units can be placed in each channel before theoutput port to provide suitable temporal delays that balance the opticalpath lengths of each channel. In this manner, a single input pulse canemerge as a set of temporally synchronized output pulses.

The PDM 400 described herein can be understood as a subset of a directedbinary tree, where each RBS corresponds to a node of the tree, and theintervening waveguides correspond to branches that connect them.

Alternatively, the PDM can be realized by a full binary tree. In thiscase, the PDM can include a cascade of RBS layers. FIG. 4B shows aschematic of a PDM 420 based on a full binary tree structure. The firstlayer includes RBS 430(1), and the second layer includes RBSs 430(2) and430(3). Two layers are shown in FIG. 4B for illustrative purposes. Eachlayer of RBSs can be split into two channels, until the desired numberof output modes is realized. This approach can use fewer RBS units (byapproximately a factor of 2 for large N), but may have a greater depthin the horizontal dimension. Depending on the footprint constraint, thePDM may be replaced by such a full-tree system.

Both approaches described above to construct the PDM 400 and 420 useglobal reconfiguration for each desired output setting. In general, eachRBS can be adjusted to change the output power in any channel. Whilethis scheme may add complexity to the control systems, it has the markedadvantage of energy efficiency, since all of the input power is used.Accordingly, these approaches are also referred to as power efficientschemes.

Alternatively, another scheme for PDM can use a fixed splitting ratiofor each RBS followed by attenuators to control the output, i.e.,equal-splitting scheme. This approach may have a simpler control systembut some of the input power may be lost during operation (e.g., at theattenuators).

Several factors may influence the decision about which scheme to use.For example, for simulations involving large squeezing levels, largenumber of modes (e.g., 5-10 modes) in realistic simulations, and largeoutput powers from the PDM, the power efficient scheme is advantageous.In another example, if simplification of the control system is desired,an equal-splitting scheme followed by variable attenuation can also beused.

Referring back to FIG. 3, the squeezer 314 is configured to generatesqueezed light with tunable squeezing. In some embodiments, the outputof the squeezer 314 includes single mode squeezed states that aredegenerate in frequency. In some embodiments, the output of the squeezer314 includes two-mode or multimode squeezed states by exploitingmultiple frequencies, polarizations, or time bins.

Various approaches can be used to generate squeezed light by thesqueezer 314. Typically, the squeezer 314 uses a paraffinic interactionin a second-order or third-order nonlinear optical medium to generatesqueezed light, i.e., spontaneous parametric down-conversion orspontaneous four-wave mixing. In some embodiments, the squeezer 314employs a dual-pump scheme to generate squeezed light via spontaneousfour-wave mixing in a resonant structure, such as a micro-ringresonator. In some embodiments, the squeezer 314 employs a single pumpscheme to generate squeezed light via spontaneous four-wave mixing in aresonant structure. In some embodiments, the spontaneous four-wavemixing can occur in a long waveguide segment (e.g., using eitherdual-pump or single pump scheme).

In some embodiments, the squeezer 314 can use spontaneous parametricdown conversion in a resonant structure (e.g. micro-ring resonator or asystem of coupled micro-ring resonators) to generate squeezed light. Insome embodiments, the spontaneous parametric down conversion can occurin a phase-matched waveguide segment, or quasi-phase matchedperiodically poled waveguide segment.

In some embodiments, in each approach of squeezed light generationdescribed herein, auxiliary structure can be included in the squeezer314 to enhance the properties of the generated squeezed light. Forexample, in resonant structures (e.g., micro-ring), it is beneficial forthe resonances that accommodate the generated squeezed light to have ahigh escape efficiency (i.e., the efficiency for the generated squeezedlight to be coupled out of the resonant structure). This can beaccomplished, for example, by strongly over-coupling the correspondingresonance mode with the resonant structure.

The squeezing factor of the output of the squeezer 314 can be adjustedby the QPU driver 340 (e.g., manually by a user or automaticallyaccording to certain protocol). In some embodiments, the input pumppower can be controlled to deliver a variable amount of optical pumppower to each squeezer 314 (e.g., via the settings of the PDM 312). Insome embodiments, for resonant squeezers, the detuning of the pump beamsaway from resonance modes in the resonator can be adjusted to change thesqueezing factor. In some embodiments, signals from the QPU driver 340are employed to actuate on-chip tunable phase shifters, which in turnimplement the above adjustments.

In some embodiments, the squeezer 314 also includes monitoring andactive feedback for stabilization (especially for resonant squeezerstructures). In these embodiments, optical signals that provideinformation on the status of each squeezer 314 are fed to a set ofmonitoring photodetectors (e.g., 315), which are in communication withthe QPU driver 340. Electrical feedback signals are then carried fromthe QPU driver 340 to the chip (i.e., components fabricated in thesubstrate 305) to actuate stabilizing changes, thereby realizing afeedback loop. In some implementations, the optical signals formonitoring can be acquired directly from the squeezer 314. This approachcan be used when, for example, the squeezer 314 includes a resonantsystem where the pump light is carried to and extracted from thesqueezer 314 in separate optical channels from the generated squeezedlight, in some implementations, the optical signals can be acquired fromoutput ports of the filter 316 as discussed below.

In some embodiments, some or all of the optical pump power for thesqueezer 314 is carried away via the same optical channel for thesqueezed light. In these embodiments, it can be useful to divert aslarge a fraction as possible (e.g., about 99% or more) of the pump lightbefore the interferometer 320. Such diversion has at least two benefits.First, the diversion can provide optical information on the squeezerstatus via the post-squeezer pump light for monitoring and locking.Second, the diversion can reduce or avoid noise generated after thesqueezer 314 from unwanted nonlinear effects driven by the pump light,such as spontaneous Raman scattering or four-wave mixing.

In the QPU 300, the diversion of the pump light for the squeeze 314 isaccomplished by the filter 316. Various types of filtering mechanismscan be employed to construct the filter 316. In some embodiments, thefilter 316 includes one or more add-drop ring filters that areconfigured to couple out pump light beams while transmitting thesqueezed light. In some embodiments, the filter 316 includes asymmetricMach-Zehnder interferometers (AMZIs). In some embodiments, the filter316 includes lattice filters (also referred to as cascaded AMZIs). Insome embodiments, the filter 316 includes Bragg grating structures. Insome embodiments, the filter 316 includes coupled ring structures. Insome embodiments, TPS units can be employed to apply biasing for thefilter 316.

FIG. 5 show schematic of a displacement module 500 that can be used toproduce displaced squeezed light for GBS, according to an embodiment.The displacement module 500 can also be used as the displacement module318 in the QPU 300 shown in FIG. 3. The displacement module 500 includesan array of RBSs 530(1) to 530(N). Each RBS 530(j) receives a firstinput beam from a corresponding squeezer 510(j) and a second input beam(i.e., the displacement beam) 520(j), where j=1, 2 . . . N. Thesqueezers 510(1) to 510(N) can collectively form the squeezer 314 inFIG. 3.

As illustrated in FIG. 5, displacement can be accomplished by mixing thedisplacement light beam, provided by a PDM (e.g., the PDM 312 in FIG.3), with squeezed light provided by the squeezer 314 (e.g., after filter316). The mixing can be performed on an RBS that is biased close to 100%transmission for the squeezed light. This can avoid adding significantloss to the squeezed light path, while allowing a variable displacementto be made according to the transmission ratio of each RBS as set by theQPU driver. The phase of the displacement beams 520 can be controlled byTPS elements on the input waveguide segments that propagate thedisplacement beams 520 before the RBSs 530.

Each RBS 530 has two output ports. One port is configured to deliver thedisplaced squeezed light toward the interferometer (e.g., 320 in FIG.3). The other port can be configured to provide signals to monitorphotodiodes for status monitoring, feedback, and stabilization.

Referring back to FIG. 3, the interferometer 320 can be implemented by anetwork of beam splitters and phase shifters (e.g., RBSs). Theinterferometer 320 can include N input ports 322 and N output ports 328so as to perform a general transformation U(N) on N input optical modes.In some embodiments, N*(N−1)/2 RBSs and N phase shifters can beinterconnected via the Reck scheme to construct the interferometer 320.In this scheme, the N*(N−1)/2 RBSs can perform SU(2) transformations andN phase shifters can perform U(1) transformations. Using RBSs toconstruct the interferometer 320 allows for low losses and a smallphysical footprint.

The QPU 300 also includes a detector (not shown in FIG. 3). In someembodiments, the detector can also be fabricated on the same platform305. In some embodiments, the detector can be fabricated on a separateplatform and can be removable from the platform 305 (e.g., via fibercoupling).

The detector in the QPU 300 is configured to perform a readout of photonnumber in each distinct optical mode that is provided by theinterferometer 320. In other words, the detector in the QPU 300 hasphoton number resolution. In some embodiments, the detector can includean array of sensors, each of which is coupled to an output port 328 ofthe interferometer 320 and generates data corresponding to the number ofphotons detected in the corresponding output port 328.

In some embodiments, the detector includes transition-edge sensor (TES)detectors. TES detectors can provide accurate photon number resolutionwith negligible dark counts and high quantum efficiency, which can bebeneficial for GBS applications. TES detectors usually operate at lowtemperatures, e.g., in the mK ranges. Therefore, the TES detectors canbe coupled to the interferometer 320 via optical fibers (instead offabricating the TES detectors on the platform 305). Data acquisition ofTES detectors can be accomplished by digitizing the voltage pulsesemerging from the detectors and performing suitable computationalanalysis to resolve the photon number in each pulse and channel. Thisdata acquisition may be triggered by an appropriate timing pulse madeavailable by the QPU control system or input pump subsystem.

In some embodiments, the detector can include any other appropriate typeof photon counting detectors, for example single photon countingdetectors such as superconducting nanowire single photon countingdetectors or single photon-sensitive avalanche photodiodes. In some suchembodiments, a suitable multiplexing scheme (e.g., spatial, temporal, ormodal multiplexing) can be used to ensure a low probability that morethan one photon is incident on any single detector element. Such amultiplexing strategy can involve some multiple number L of detectors,with being a multiple of N and large enough to suppress multi-photoncounting events per detector. To ensure that the detectors only detectdesired photons, the optical fibers connected to the detectors caninclude multiple stages of filtering.

The GBS apparatus described herein with reference to FIGS. 1-5 above canbe configured such that the output samples (i.e., statistics of photonnumbers in each optical mode at the output) contain useful information.A wide array of problems can be encoded into GBS by varying the inputsqueezing and/or N-mode linear interferometer settings. These problemscan be graph-based or combinatorial optimization-based, and have anumber of practical applications including social network optimizationand chemistry. Output samples from a suitably programmed GBS device canthen be used to provide candidate solutions to encoded problems. Thesesamples may be processed by a classical computer, including as part of ahybrid classical-quantum algorithm. The result can further lead to areconfiguration/update of the GBS device as part of the continuingalgorithm.

FIG. 6 shows a block diagram to illustrate a method 600 of operating aGBS device, according to an embodiment. The method 100 includesembedding an application 610 into a GBS system 620 (e.g., any of the GBSdevices illustrated in and/or described with reference to FIGS. 1-5).For example, the squeezing factor and/or the phase settings of theinterferometer in the GBS system can be configured to solve a particularproblem (e.g., graph or vibronic spectra as discussed below). The output630 of the GBS system 620 is then sent to a classical computer 640 (orsimply a processor) for post-processing (e.g., analysis of the photondistribution in each optical mode) so as to retrieve solutions 650 ofthe particular problem embedded in the GBS system 620. In addition, theprocessed information from the classical computer 640 can be sent backto the application 610 to improve the embedding (e.g., adjusting theinput optical modes or phase setting of the interferometer). Exemplaryapplications that can be embedded in the GBS system 620 includeidentifying highly connected influencers on a social network, improvingthe efficiency of solar cells, or reducing the time between when acustomer places and receives an online order.

The GBS apparatus described herein can also allow interactions withusers, such as setting parameters of the state generation and linearoptical interferometer components. To this end, the hardware to executea GBS-based application can be programmable, i.e., the squeezingparameters, coherent displacement settings (if applicable), andinterferometer matrix can be specified and actuated accordingly.

FIG. 7 shows a schematic of a GBS system 700 that provide user access tothe setting of the quantum hardware, according to an embodiment. In theGBS system 700, the hardware elements responsible for input statepreparation, interferometer transformation, and output state detection,are henceforth referred to as the quantum hardware (QH) 750 (e.g.,similar to the QH 160 in FIG. 1). Parameter setting of QH can beaccomplished with a set of classical computers and associatedcommunication interfaces. More specifically, a master unit 730 isdedicated to running, monitoring, and receiving results from the QH 750,via a set of control systems 740. Clients can access the master unit 730(and thus the QH 750) via a user interface 710. In some embodiments, theuser interface 710 is locally connected to the master unit 730. In someembodiments, clients can access the master unit 730 via the internet(i.e., the user interface 710 includes a network interface). In eithercase, instructions can be passed to the hardware via an applicationprogram interface (API). In the internet-access model (also referred toas the cloud access model), clients can communicate with a server 720via secure web link, which, in turn, communicates with the master unit730.

As described herein, GBS apparatus and systems as illustrated in FIGS.1-7 can be used to solve a wide range of problems. Two examples, graphproblems and vibronic spectra, are discussed with more details below forillustrative purposes.

Solving problems on graphs is of both theoretical and practical interestdue to its applications in computer vision, social networks, andfinance. For example, algorithms for graph problems can help identifyhighly connected influencers on a social network, improve the efficiencyof solar cells, and reduce the time between when a customer places andreceives an online order. Graph problems may be specified through agraph's adjacency matrix, which can be encoded by varying inputsqueezing and N-mode linear interferometer settings. In this encoding,each vertex of the graph corresponds to a mode of the device/system inGBS. More details can be found in, for example, Kamil Bradler,Pierre-Luc Dallaire-Demers, Patrick Rebentrost, Daiqin Su, and ChristianWeedbrook. Gaussian boson sampling for perfect matchings of arbitrarygraphs, Phys. Rev. A, vol. 98, page 032310, September 2018, which isincorporated herein in its entirety.

FIG. 8 shows a block diagram illustrating a method 800 of solving graphproblems using GBS devices described herein, according to an embodiment.In the method 800, a graph problem 810 is solved using algorithms andheuristics 820 to reach solutions 830, and a GBS system 850 can be usedto improve the solution. More specifically, the graph problem 810 can befirst encoded into a GBS system (at 840). Samples are then taken fromthe GBS system 850 (e.g., any of the GBS system illustrated in anddescribed with reference to FIGS. 1-7) and processed within algorithmsand heuristics 820 to help solve graph problems. In some embodiments,the processing include iterative updating of the GBS device based on itsoutput. The GBS model of computation can be adapted within algorithmsfor finding dense subgraphs and maximum cliques, calculating graphmatchings, graph isomorphism, and graph similarity, which are describedwith more details below.

Identifying subgraphs with high connectivity is a relevant problem inbioinformatics, drug design, data mining, finance, and communitydetection in social networks. Unfortunately, the computational resourcesrequired to find these subgraphs grows rapidly with the size of thegraphs. GBS can be employed to randomly generate highly-connectedsubgraphs with high probability. This feature allows GBS to improveexisting classical algorithms that rely on random searches over thespace of possible subgraphs.

Classical algorithms often proceed with a mixture of global explorationof the problem space and local searching of candidate solutions,exploiting local structure. Both exploration and local search caninvolve stochastic elements. For example, choosing a random subgraph asa starting point, or locally searching to build a bigger clique. Thesestochastic elements are often biased towards selecting highly-connectedsubgraphs, with the objective of finding improved candidate solutions. Asuitably encoded GBS system can then be used as a stochastic source thatselects highly-connected subgraphs.

The graph isomorphism problem is a decision problem concerning whethertwo isospectral graphs (i.e., graphs with the same eigenspectrum) areisomorphic, i.e., whether they are related to each other by a mererelabeling of their vertices. A major open question is whether thereexists a polynomial-time algorithm that can determine whether two graphsare isomorphic. In fact, graph isomorphism is likely to belong to theclass of NF-intermediate computational problems. A further generalizedquestion concerns how two non-isomorphic graphs (including graphs ondifferent numbers of vertices) are related to each other, i.e., thegraph similarity problem. Graph isomorphism, but mainly graphsimilarity, offers solutions to difficult problems such as computervision, detecting unusual activities (e.g., fraudulent withdrawals) onfinancial transaction networks, and the determination of properties offamilies of molecules e.g., in pharmaceutic, genetics, or biochemistry)without the need for their synthesis.

GBS can be leveraged to yield a complete set of graph invariants, withtwo graphs being isomorphic if and only if these graph invariants areequal. To this end, the link between graph isomorphism and GBS is firstexplained. It can be understood by considering cyclic permutations ofphoton events from GBS. One useful concept is the orbit representative,which acts as a seed event to describe all permutationally equivalentevents forming orbits. For example: the orbit of (0, 0, 1, 2) describesthe possible events (0, 1, 2, 0), (2, 1, 0, 0), and so forth. Theseevents can be associated with output samples (also referred to as clickpatterns) from a GBS device, with each orbit having a corresponding GBSprobability. The orbit probability can be found for two separate graphsand compared. If the orbit probabilities differ, it is sufficient toconclude that the two graphs are not isomorphic.

An implementation of this scheme can proceed by encoding two graphs on aGBS device as discussed above and evaluating the probability of a givenorbit by taking multiple samples from the GBS device. In someembodiments, the graphs can be encoded into the GBS device sequentially,i.e., evaluating the orbit probability for one graph, and thenevaluating for the other. In some embodiments, the graphs can be encodedinto the GBS device alternately, i.e., taking samples from one graph andthen the other, updating their orbit probabilities continuously. Enoughsamples can be taken such that their orbit probabilities do not overlapwhen experimental uncertainties are also considered. This can be helpfulto conclude that two graphs are non-isomorphic.

To describe the application of GBS for the graph similarity problem, itis also helpful to link GBS orbits with two concepts in graph theory:k-matching and the matching polynomial. A k-matching in a simple (i.e.,undirected and loopless) graph on 2M vertices is the number of kindependent (disjoint) edges of the graph. By definition, 1≤k≤M. Thek-matchings are the coefficients of the graph matching polynomial.Calculating the matching polynomial can be a challenging computationalproblem that becomes intractable for traditional computers when M issufficiently large. In fact, there is no known efficient traditionalalgorithm to even approximate the matching polynomial, except forcertain special classes of graphs.

A GBS device can be used to estimate the matching polynomial via the useof the GBS polynomial. The GBS polynomial is of the same order as thematching polynomial and its coefficients are related, but not identical,to the graph's k-matchings. Some of the terms (also referred to ascoefficients) of the GBS polynomials can be efficiently estimated usingthe GBS device and, moreover, the GBS polynomial is more suitable forthe graph isomorphism or similarity problem than using the informationone can get from the graphs' matching polynomials.

The GBS polynomials can be extracted from the output of a GBS device asfollows. An output click pattern of a (IBS device is referred to ascollision-free if at most one photon is detected at each output mode.When a graph is encoded in a GBS device, it can be shown that theprobability of observing a collision-free orbit with 2 k photons isequal to the k-th coefficient of the GBS polynomial. The importantaspect of sampling orbits with 2 k photons is that this is tractable forsome k. This is because the number of collision-free orbits grows merelypolynomially with the encoded graph size 2M. Since the orbitprobabilities are graph invariants, they provide an efficient way forGBS to decide whether two graphs are isomorphic.

The extension to graph similarity is given by considering the full rangeof possible orbits given with a fixed total photon number. This takesthe problem out of the no collision regime and accordingly one has tocorrespondingly generalize the GBS polynomial insight. Higher orderprobability distributions include a combination of different orbitprobabilities. These coarse-grained probability distributions arecoefficients of another GBS polynomial, which corresponds to an extendedgraph obtained by tensoring the original graph with a complete graphwith self-loops.

There are three aspects to the above process. First, the morecoarse-grained the probability distributions, the easier it is to use aGBS device to sample from them. Second, the tensoring procedure canamplify the differences between two graphs. It is the comparison of theextended GBS polynomial coefficients that makes it so suitable for thegraph similarity problem. Taking coarse-grained orbits with higherphoton numbers can obtain more refined information about the differencebetween two graphs. Finally, these insights lead to a comparison of thematching polynomials (as if one has access to them) and the GBSpolynomial and in alt cases it turns out that the GBS polynomial revealsthe dissimilarity of two graphs for a lower number of detected photons.This last point suggests that the GBS polynomial can be more powerfulthan the matching polynomial for this type of problems and this hasramifications for the experimental implementations of the algorithm.

A GBS scheme that can be even more suitable for the graph similarityproblem includes initial uniformly displaced squeezed states enteringthe linear interferometer encoding the graph. The considered regime canbe both collision-containing or collision-free. The comparison of thedetected click patterns and their coarse-grained versions leads to ameasure of similarity for the encoded graphs. In general, with orwithout a displacement, evaluating the orbit distribution through GBSallows a user to associate an encoded graph with a feature vectorpopulated with the probabilities of orbits. This association can beunderstood as an embedding of a graph into real space using GBS, whichcan be used to formulate a graph kernel. Graph kernels and featurevectors can be used within the scope of machine learning, such as forclassification and regression. This concept can be extended to a form ofclassification or clustering of graphs, using methods such as supportvector machines and k-means clustering.

Based on the descriptions above, a GBS-based solution to a graph problem(e.g., graph isomorphism or similarity problem) can be carried out asfollows. First, a graph is made GBS-encodable by converting the graphinto a Gaussian covariance matrix. The Gaussian covariance matrix isused to find the squeezing parameters and the unitary angle and phasevalues for the linear interferometer in the GBS device. Optimized vacuumdisplacement values are found for the displacement-based version of thesetup. Then, the statistics of the click patterns is collected from thedetectors in the GBS device. The coarse-grained statistics are thenobtained by classical post-processing. Orbit probabilities are estimatedin the collision or collision-free regime. They are interpreted as thecoefficients of the GBS polynomial of the graph or its tensoredextensions.

Next, for the graph isomorphism problem, the difference in the estimatedGBS polynomial coefficients is used to conclude that two graphs are notisomorphic. For the graph similarity problem, the relevant estimated GBSpolynomial coefficients are associated with the feature vectors andprocessed in the machine learning layer to find how similar two graphsare. For example, the feature vectors are processed by a classicalcomputer running a machine learning algorithm, such as a neural networkor a kernel machine, among others. Finally, the decision problem or thesimilarity measure can be reported to the user.

Effectively modeling molecular vibronic spectra is useful forunderstanding and engineering chemical reactions for both academicresearch and industrial purposes. Accurately predicting the structure ofmolecular vibronic spectra can be a computationally intensive task andusually involves prohibitively large computational resources as themolecules to be targeted grow in size and complexity. A strategy toperform these calculations efficiently would therefore represent animportant step forward for chemical simulation.

A GBS system enables such a strategy by exploiting a connection betweenthe dynamics of vibrational modes and quantum optics. For a moleculewith n vibrational modes, the GBS strategy involves the preparation of ndisplaced squeezed states, an n×n linear optical interferometer, andphoton number resolved detection in the output modes. In all cases abovethe associated GBS algorithm scales better than any known classicalalgorithm in computational resources, and/or the number of calculationsteps, with the size of the problem.

In a GBS-based approach to solve vibronic spectra problems (e.g., thesystem shown in FIG. 7), a client can first communicate with a servervia a secure web link, using an appropriate graphical or command-lineuser interface, or custom application program interface (API). Theinformation transmitted to the server by the client describes themolecular structures to be simulated. These molecular structures areencoded as physical parameters (e.g., optical powers, beam splitterratios, and phases) by an interpreter, which can include softwarerunning on a conventional computer. The interpreter passes thesephysical parameters to a control system, which includes auxiliarydrivers that directly communicate with the quantum hardware. The quantumhardware includes the optical chips and photon counting detectors. Theoutputs from these detectors are fed to software running on a decoder,which parses the photon counting distribution into a usablerepresentation of the vibronic spectra. This information is passed backto the server, which communicates the results of the calculations backto the client.

In some embodiments, the client access model shown in FIG. 7 can use themolecular coding scheme described in Huh et al., Boson sampling formolecular vibronic spectra, Nature Photonics 9, 615 (2015), which ishereby incorporated by reference in its entirety. The interpreter usesthis encoding scheme to translate the vibronic transition to besimulated into a set of parameters, including beam splitter ratios andphases for the interferometer, squeezing factors and displacements to beimplemented by the squeezer/displacer. In systems described herein,these parameters can be controlled on-chip by phases in each module.Thus, the interpreter ultimately instructs the control system whatphases to set in each module. The control system converts this to a setof voltages to apply to the phase shifters.

For the decoding, the photon counts measured by the detectors can formprecisely the probability distribution that makes up the set ofFranck-Condon (FC) factors that can be used to reconstruct the vibronicspectra. So, the decoder can take this photon counting distribution,compile it into a set of FC factors, and report the result back to theclient via the server.

In addition to solving graph problems and modeling vibronic spectra, GBSsystems described herein can also be used as point processes. Withoutbeing bound by any particular theory or mode of operation, pointprocesses are statistical models for generating and interpreting randompoint patterns that might appear in nature, technology, and humanaffairs. Matrix function point processes are a class of point processesin which the probability of observing a specific point pattern dependson matrix functions such as determinants and permanents. Determinantalpoint processes can be applied to various problems in machine learningincluding text summarization, human pose estimation and news threading.Point processes that depend on permanents or hafnians, however, have notbeen broadly used due to the computational difficulties associated withthe calculation of these matrix functions on classical computers.

A GBS machine can be considered as a point process that generates randompoint patterns in the form of photons in certain optical modes. Theprobabilities of patterns generated by GBS are proportional to thehafnian of the matrix that specifies the GBS settings. Accordingly, aGBS machine can be programmed to generate point patterns that inheritthe typical features of hafnian and permanental point processes. As aresult, GBS overcomes the difficulties associated with the applicationof hafnian and permanental point processes on classical computers. Twoexamples of the GBS point process, among others, are provided in thefollowing.

One feature of the GBS point process is the higher probability of pointpatterns that are locally clustered. This intrinsic clustering propertyof GBS can be used in tasks such as portfolio optimization and machinelearning. One aspect of the portfolio optimization is to diversifyassets in order to avoid having highly correlated stocks in oneportfolio. The GBS machine can be programmed with the stock covariancematrices obtained based on the time correlation of stock properties suchas market prices. This GBS point process is able to detect highlycorrelated stocks and thus helps to provide well-diversified and stableportfolios. The GBS point process can also be combined with classicalalgorithms to increase the efficiency of classical machine learningclustering methods. In particular, the GBS point process can be used toprovide more realistic cluster centers used to initiate classicalclustering algorithms such as k-means.

While various embodiments have been described and illustrated herein, avariety of other means and/or structures for performing the functionand/or obtaining the results and/or one or more of the advantagesdescribed herein, and each of such variations and/or modifications arepossible. More generally, all parameters, dimensions, materials, andconfigurations described herein are meant to be examples and that theactual parameters, dimensions, materials, and/or configurations willdepend upon the specific application or applications for which thedisclosure is used. It is to be understood that the foregoingembodiments are presented by way of example only and that otherembodiments may be practiced otherwise than as specifically describedand claimed. Embodiments of the present disclosure are directed to eachindividual feature, system, article, material, kit, and/or methoddescribed herein. In addition, any combination of two or more suchfeatures, systems, articles, materials, kits, and/or methods, if suchfeatures, systems, articles, materials, kits, and/or methods are notmutually inconsistent, is included within the inventive scope of thepresent disclosure.

Also, various concepts may be embodied as one or more methods, of whichan example has been provided. The acts performed as part of the methodmay be ordered in any suitable way. Accordingly, embodiments may beconstructed in which acts are performed in an order different thanillustrated, which may include performing some acts simultaneously, eventhough shown as sequential acts in illustrative embodiments.

All definitions, as defined and used herein, should be understood tocontrol over dictionary definitions, definitions in documentsincorporated by reference, and/or ordinary meanings of the definedterms.

As used herein, a “module” can be, for example, any assembly and/or setof operatively-coupled electrical components associated with performinga specific function, and can include, for example, a memory, aprocessor, electrical traces, optical connectors, software (stored andexecuting in hardware) and/or the like.

The indefinite articles “a” and “an,” as used herein in thespecification and in the claims, unless clearly indicated to thecontrary, should be understood to mean “at least one.”

The phrase “and/or,” as used herein in the specification and in theclaims, should be understood to mean “either or both” of the elements soconjoined, i.e., elements that are conjunctively present in some casesand disjunctively present in other cases. Multiple elements listed with“and/or” should be construed in the same fashion, i.e., “one or more” ofthe elements so conjoined. Other elements may optionally be presentother than the elements specifically identified by the “and/or” clause,whether related or unrelated to those elements specifically identified.Thus, as a non-limiting example, a reference to “A and/or B”, when usedin conjunction with open-ended language such as “comprising” can refer,in one embodiment, to A only (optionally including elements other thanB); in another embodiment, to B only (optionally including elementsother than A); in yet another embodiment, to both A and B (optionallyincluding other elements); etc.

As used herein in the specification and in the claims, “or” should beunderstood to have the same meaning as “and/or” as defined above. Forexample, when separating items in a list, “or” or “and/or” shall beinterpreted as being inclusive, i.e., the inclusion of at least one, butalso including more than one, of a number or list of elements, and,optionally, additional unlisted items. Only terms clearly indicated tothe contrary, such as “only one of” or “exactly one of,” or, when usedin the claims, “consisting of,” will refer to the inclusion of exactlyone element of a number or list of elements. In general, the term “or”as used herein shall only be interpreted as indicating exclusivealternatives (i.e. “one or the other but not both”) when preceded byterms of exclusivity, such as “either,” “one of,” “only one of,” or“exactly one of” “Consisting essentially of,” when used in the claims,shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase “atleast one,” in reference to a list of one or more elements, should beunderstood to mean at least one element selected from any one or more ofthe elements in the list of elements, but not necessarily including atleast one of each and every element specifically listed within the listof elements and not excluding any combinations of elements in the listof elements. This definition also allows that elements may optionally bepresent other than the elements specifically identified within the listof elements to which the phrase “at least one” refers, whether relatedor unrelated to those elements specifically identified. Thus, as anon-limiting example, “at least one of A and B” (or, equivalently, “atleast one of A or B,” or, equivalently “at least one of A and/or B”) canrefer, in one embodiment, to at least one, optionally including morethan one, A, with no B present (and optionally including elements otherthan B); in another embodiment, to at least one, optionally includingmore than one, B, with no A present (and optionally including elementsother than A); in yet another embodiment, to at least one, optionallyincluding more than one, A, and at least one, optionally including morethan one, B (and optionally including other elements); etc.

In the claims, as well as in the specification above, all transitionalphrases such as “comprising,” “including,” “carrying,” “having,”“containing,” “involving,” “holding,” “composed of,” and the like are tobe understood to be open-ended, i.e., to mean including but not limitedto. Only the transitional phrases “consisting of” and “consistingessentially of” shall be closed or semi-closed transitional phrases,respectively, as set forth in the United States Patent Office Manual ofPatent Examining Procedures, Section 2111.03.

What is claimed is:
 1. An apparatus, comprising: a light sourceconfigured to provide a plurality of input optical modes in a squeezedstate of light; a network of interconnected reconfigurable beamsplitters (RBSs) in optical communication with the light source, thenetwork of interconnected RBSs configured to perform a unitarytransformation of the plurality of input optical modes to generate aplurality of output optical modes; an array of photon counting detectorsin optical communication with the network of interconnected RBSs, thearray of photon counting detectors configured to measure a number ofphotons in each output optical mode of the plurality of the outputoptical modes after the unitary transformation; and a controlleroperatively coupled to the light source and the network ofinterconnected RBSs, the controller configured to control at least oneof a squeezing factor of the squeezed state of light, an angle of theunitary transformation, or a phase of the unitary transformation.
 2. Theapparatus of claim 1, wherein the light source is configured to providethe plurality of optical modes in a displaced squeezed state.
 3. Theapparatus of claim 1, wherein the light source includes: a powerdivision module (PDM) to receive an input optical beam and divide theinput optical beam into 2N PDM beams, where N is a positive integer; anda squeezer-displacer module (SIM), in optical communication with thePDM, to receive the 2N PDM beams and generate the plurality of inputoptical modes having N squeezed light beams.
 4. The apparatus of claim1, wherein the network of interconnected RBSs includes an array of inputports to guide the plurality of input optical modes into the network ofinterconnected RBSs and an array of output ports to send the pluralityof output optical modes into the array of photon counting detectors. 5.The apparatus of claim 1, wherein the array of photon counting detectorsincludes an array of transition-edge superconducting detectors.
 6. Theapparatus of claim 1, further comprising: a substrate including at leastone of silicon, silicon oxide, or silicon nitride, the light source andthe network of interconnected RBSs being fabricated in the substrate. 7.The apparatus of claim 1, wherein the light source is configured toprovide the plurality of input optical modes representing an adjacencymatrix of a graph, each input optical mode in the plurality of inputoptical modes representing a vertex of the graph.
 8. The apparatus ofclaim 1, wherein the light source is configured to provide the pluralityof input optical modes representing an adjacency matrix of a graph, eachinput optical mode in the plurality of input optical modes representinga vertex of the graph, the apparatus further comprising: a processoroperatively coupled to the array of photon counting detectors, theprocessor configured to find dense subgraphs within the graph based onthe number of photons in each output optical mode of the plurality ofthe output optical modes after the unitary transformation.
 9. Theapparatus of claim 1, wherein the light source is configured to providethe plurality of input optical modes representing an adjacency matrix ofa graph, each optical mode in the plurality of input optical modesrepresenting a vertex of the graph, the apparatus further comprising: aprocessor operatively coupled to the array of photon counting detectors,the processor configured to calculate graph invariants for determininggraph isomorphism and GBS polynomial for graph isomorphism andsimilarity based on the number of photons in each output optical mode ofthe plurality of the output optical modes after the unitarytransformation.
 10. The apparatus of claim 1, wherein the light sourceis configured to provide the plurality of input optical modesrepresenting an adjacency matrix of a graph, each input optical mode inthe plurality of optical modes representing a vertex of the graph, theapparatus further comprising: a processor operatively coupled to thearray of photon counting detectors, the processor configured to evaluategraph isomorphism based on the number of photons in each output opticalmode of the plurality of the output optical modes after the unitarytransformation.
 11. The apparatus of claim 1, further comprising: a userinterface operatively coupled to the light source and the network ofinterconnected RBSs, the user interface configured to receiveinformation about a molecular structure provided by a user; and aninterpreter, operably coupled to the user interface, to generate acontrol signal based on the information about the molecular structure,the controller being configured to cause the light source to generatethe plurality of input optical modes in response to the control signal.12. The apparatus of claim 1, further comprising: a decoding unitoperably coupled to the array of photon counting detectors andconfigured to transform an array of detection signals acquired by thearray of photon counting detectors into a representation of molecularvibronic spectrum.